Giovanni PISTONE - Collegio Carlo Alberto, Torino - NON-PARAMETRIC INFORMATION GEOMETRY WITH DERIVATIVES

Non-parametric Information Geometry according to a series of papers starting
with [6] consists of a manifold on the set of positive densities of a measure space.
The manifold is modeled on the Banach space of exponentially integrable random
variables. In a more recent presentation [4] the relevant structure is described a
Banach bundle of couples (p, u) where p is a positive density and u is a random
variable such that Ep(u) = 0. Each connected component of the base manifold,
consisting of densities which are connected by an open exponential family, is fully
described in [7]. Other methods for dealing with the infinite-dimensional geometry
of probabilities are available, in particular [1]. The main limitation of this approach
is the inability to deal with properties of the statistical models depending on the
structure of the sample space where the densities are denes e.g., the smoothness. In
the framework of Gaussian spaces [2] it is actually possible to study such properties
while retaining the same bundle structure. Preliminary results have been published
in [3, 5] and further research is in progress. An example of application is the study
of Hyvarinen divergence [2]

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